Periodic Functions and
Symmetry
Symmetry
Section 2.1
A
periodic function is a function that repeats
periodic
itself.
itself.
Even Functions
Even
If
If
f(-x) = f(x) for every value in its domain,
f(-x)
then the function is an even function.
then
Even
axi
The Law of Sines and
Law of Cosines
Sections 4.1 and
4.3
Law of Sines
sin A sin B sin C
=
=
a
b
c
c
A
B
a
C
Law of Cosines
a = b + c 2bc cos A
2
2
2
b = a + c 2ac cos B
2
2
2
c = a + b 2ab cos C
2
2
2
c
A
B
a
C
Joke Time
Why are toucans always in debt?
Be
RatesofChange
Section3.7
AverageRateofChange
AverageRateofChange
Letfbeafunction.Theaveragerateofchangeof
f(x)withrespecttoxasxchangesfromatobisthe
value
change in f ( x) f (b) f (a )
=
change in x
ba
Theaveragerateofchange=slopeofsecantline
joining(a,f
Proving Identities
Proving
Section 3.6
Strategies for Proving Identities
Strategies
1.
2.
3.
Know the fundamental identities and
Know
look for ways to apply them.
look
Write all expressions in terms sine
Write
and cosine.
and
Work with one side of the ide
Inverse Functions
Inverse
Section 3.6
Graphs of Inverse Relations
Graphs
Let f be a function. If (a, b) is a point o the
Let
(a,
graph of f, then (b, a) is a point o the graph
of its inverse.
of
The graph of the inverse of f is a reflection of
The
the g
Operations on
Operations
Functions
Functions
Section3.5
Sum, Difference, Product, and
Quotient of Functions
Quotient
( f + g )( x) = f ( x) + g ( x)
( f g )( x) = f ( x) g ( x)
( fg )( x) = f ( x) g ( x)
f
f ( x)
( x) =
g ( x)
g
Composite Functions
Compo
Fundamental Identities
Fundamental
Section 3.4
Reciprocal Identities
Reciprocal
1
csc =
sin
1
cot =
tan
1
sec =
cos
Ratio Identities
Ratio
sin
tan =
cos
cos
cot =
sin
Pythagorean Identities
Pythagorean
sin + cos = 1
2
2
1 + cot = csc
2
2
1 + tan =
Quadratic Functions
Quadratic
Section 3.3
Parabolas
Parabolas
The rule of a quadratic function is a
polynomial of degree 2. The shape is a
parabola.
vertex
x-intercepts
Forms of a Quadratic Function
Forms
Transformation form:
Polynomial form:
x-Interc
Applications
Section 3.3
The angle measured clockwise from
north to the line of travel is the course of
the plane or ship.
N
Course
o
ne
Li
h
sig
f
t
110o
Lin
eo
f tr
av
el
The clockwise angle from north to the
line of sight to a point of reference is
c
Angles of Elevation
and Depression
Section 3.2
angle of elevation an angle
formed by a horizontal line
and the line of sight to an
object above the level of the
horizontal.
angle of elevation
angle of depression an
angle formed by a horizontal
line and th
Graphs of the Secant and
Cosecant Function
Cosecant
Section 2.7
y = sec x
sec
3
y = sec x
2
1
A
-2
2
-1
-2
4
y = csc x
csc
3
y = csc x
2
1
A
-2
2
-1
-2
4
For y = a sec b(x +c) + d ; y = a cot b(x +c) + d
a vertically stretches or shrinks the graph.
The
Graphs of the Tangent
and Cotangent Function
and
Section 2.6
y = tan x
tan
f( x) = tan( x)
3
2
1
-2
2
-1
-2
-3
4
y = cot x
cot
y = cot x
3
2
1
-2
2
-1
-2
4
For y = a tan b(x +c) + d ; y = a cot b(x +c) + d
a vertically stretches or shrinks the graph.
Th
Polynomial Functions
Polynomial
Section 4.1
Defn. of a Polynomial Function
Defn.
A polynomial function is a function whose rule
polynomial
is given by a polynomial
is
f ( x) = an x n + an 1 x n 1 + L + a1 x + a 0
an , an 1 ,K , a1 , a0
where
where
with
Real Zeros
Section4.2
The Rational Zero Test
The
r
Ifarationalnumberisazeroofthepolynomial
s
function
then 0
f ( x) = an x n + L + a1 x + a
risafactoroftheconstanttermanda0
sisafactoroftheleadingcoefficient
an
Bounds Test
Bounds
Letf(x)beapolynomialwithp
HONORS COURSE PORTFOLIO
Instructions: For each Honors Component Option mark the Content Criteria that will be met in your product.
HONORS COMPONENT OPTION #1
Evidence of extended reading assignments that connect with specified curriculum
Content Criteria:
Inverse Trig. Functions
Sections 6.2 and 6.3
Inverse Trig. Functions
None of the 6 basic trig. functions
has an inverse unless you restrict
their domains.
1
y = Sin x iff
y
2
2
sin y = x and
1
y = Cos x iff
0 y
cos y = x and
Function
y = Arcsin x
y = Ar
Inverse Functions
Inverse
Section 6.1
Reflective Property of
Inverse Functions
Inverse
The graph of f contains the point
(a, b) iff the graph of f -1 contains
the point (b, a)
The Existence of an Inverse
Function
Function
A function has an inverse iff it
Half-Angle Identities
Half-Angle
Section 5.5
Half-Angle Identities
Half-Angle
x
1 cos x
sin =
2
2
x
1 + cos x
cos =
2
2
x
1 cos x
tan =
2
1 + cos x
x 1 cos x
tan =
2
sin x
x
sin x
tan =
2 1 + cos x
Joke Time
Joke
What did E.T.s mom say when he got
hom
Common and Natural
Logarithmic Functions
Sections 5.4, 5.5 and
5.6
Common logarithms:
log v = u iff
10 = v
u
Natural logarithms:
ln v = u iff
e =v
u
Basic Properties of
Logarithms
1. log v and ln v are defined only when
v > 0.
2. log 1 = 0 and log 10
Double-Angle Identities
Section5.4
Double Angle Identities
Double
sin 2 x = 2sin x cos x
cos 2 x = cos x sin x
2
cos 2 x = 2 cos x 1
2
cos 2 x = 1 2sin x
2
2
2 tan x
tan 2 x =
2
1 tan x
Joke Time
Joke
Whydobeeshavestickyhair?
Becausetheyhavehoneycombs!
Wh
Applications of Exponential
Functions
Functions
Section 5.3
Compound Interest
Compound
If
If
P dollars is invested at interest rate r
(expressed as a decimal) per time period t,
then A is the amount after t periods.
A = P (1 + r )
t
Continuous Compoundin
Exponential Functions
Exponential
Section 5.2
Graph of Exponential Function
Graph
a>0
graph is above x-axis
y-intercept is 1
f(x) is increasing
f(x) approaches the negative x-axis as x
approaches
0<a<1
graph is above x-axis
y-intercept is 1
f(x) is dec
Exponential and
Logarithmic Functions
Logarithmic
Section 5.1
nth Roots
nth
Let c be a real number and n a positive integer:
The nth root of c is denoted by either of the
nth
symbols
symbols
n
c
or c
1
n
and is defined to be
solution of x = c when n is od
VectorsinthePlane
VectorsinthePlane
Section4.7
Quantitiesthathavemagnitude(length)and
directionarecalledvectors.
u
Samemagnitude,
v
Differentdirection
u=v
Same
direction,
Different
magnitude,
Different
Differentdirection
Ifavectorisplacedonastandardcoordi
Area of Triangle and
Herons Formula
Herons
Sections 4.5 and 4.6
B
h
a
A
C
h
sin C =
a
h = a sin C
1
K = bh
2
1
K = ba sin C
2
Area of a Triangle
Area
The area K of a triangle is
1
K = ab sin C
2
1
K = bc sin A
2
1
K = ac sin B
2
This formula is used whe
Rational Functions
Section 4.4
Domain of Rational Functions
The domain of a artional function is the
set of all real numbers that are not
zeros of its denominator.
Intercepts of Rational
Functions
If f has a y-intercept, it occurs at f(0).
The x-intercep